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Pattern recognition with vector hits
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2012 JINST 7 C09002
(http://iopscience.iop.org/1748-0221/7/09/C09002)
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P UBLISHED BY IOP P UBLISHING FOR S ISSA M EDIALAB
R ECEIVED: June 8, 2012
ACCEPTED: August 22, 2012
P UBLISHED: September 12, 2012
WIT2012 — W ORKSHOP
3–5 M AY 2012,
INFN P ISA , I TALY
ON I NTELLIGENT TRACKERS ,
R. Frühwirth1
Institute of High Energy Physics, Austrian Academy of Sciences,
Nikolsdorfer Gasse 18, 1050 Wien, Austria
E-mail: [email protected]
A BSTRACT: Trackers at the future high-luminosity LHC, designed to have triggering capability,
will feature layers of stacked modules with a small stack separation. This will allow the reconstruction of track stubs or vector hits with position and direction information, but lacking precise
curvature information. This opens up new possibilities for track finding, online and offline. Two
track finding methods, the Kalman filter and the convergent Hough transform are studied in this
context. Results from a simplified fast simulation are presented. It is shown that the performance
of the methods depends to a large extent on the size of the stack separation. We conclude that the
detector design and the choice of the track finding algorithm(s) are strongly coupled and should
proceed conjointly.
K EYWORDS : Pattern recognition, cluster finding, calibration and fitting methods; Particle tracking
detectors
1 Corresponding
author.
c 2012 IOP Publishing Ltd and Sissa Medialab srl
doi:10.1088/1748-0221/7/09/C09002
2012 JINST 7 C09002
Pattern recognition with vector hits
Contents
Introduction
1
2
The “Long Barrel”
1
3
Vector hits
2
4
Track finding with Kalman filter
3
5
Track finding with convergent Hough transform
6
6
Discussion and outlook
6
1
Introduction
The next generation of trackers at the high-luminosity LHC will be designed to have trigger capabilities already at the lowest, fastest level. This will most likely be achieved by the concept of
stacked modules consisting of two closely spaced silicon strip or pixel sensors. Hits from tracks
with low transverse momentum can then be rejected by requiring a coincidence that is consistent
with a track with high transverse momentum (see figure 1 in [1]). The two hits from the stacked
module can be combined to a track stub or vector hit, which carries both position and direction
information. In addition, if the stack separation is not too small, a reasonable estimate of the
transverse momentum can be obtained by using the beam profile in the transverse plane.
This contribution investigates the impact of such a design on the pattern recognition capability
of the tracker. In particular, we study the performance of two algorithms: track following with the
Kalman filter (KF) [2, 3], and a local or convergent Hough transform (CHT) [4, 5]. This should
by no means be construed as a claim that these particular methods are best suited to the problem
at hand. Rather, it is intended to illustrate how the layout of the detector and in particular the size
of the stack separation influences the performance of certain algorithms. Obviously, a much larger
class of methods has to be studied if one aims at an optimal combination of layout and track finding
algorithm. The conclusions of this study are therefore very preliminary.
2
The “Long Barrel”
The results presented in the subsequent sections are based on events simulated in a particular geometry, which is a slight modification of the “long-barrel” detector concept as shown in ([1], figure 2).
The layout can be summarized as follows:
• Six stacked layers at r = 0.30, 0.35, 0.50, 0.60, 0.95, 1.05 m;
• Stack separation ∆r: between 2 mm and 8 mm;
–1–
2012 JINST 7 C09002
1
Table 1. Summary of the stack separations per layer (in mm) in the five simulation runs. Ln refers to the
n-th layer.
Run
A
B
C
D
E
L1
2
2
2
4
8
L2
2
2
2
4
8
L3
2
3
4
6
8
L4
2
3
4
6
8
L5
2
4
6
8
8
L6
2
4
6
8
8
Method
KF
KF
KF, CHT
CHT
CHT
• Material: 6 × 2% = 12% of a radiation length;
• Magnetic field: Bx = By = 0, Bz = 4 T.
The event simulation was done by a fast simulation program, without secondary interactions.
This is a serious limitation, which can be redressed only by passing to a full simulation using
GEANT4. This, however, is beyond the scope of the present work. Another simplifications concerns the generation of the detector hits, which was done by placing the hit in the pixel center.
Both methods were studied using different assumptions about the stack separation, summarized in table 1. Each of the five runs consisted of 500 simulated events with the following configuration:
• 1000 “primary” tracks, with ϕ uniform in [0, 2π], z of track uniform in [−100 mm, 100 mm],
η uniform in [−1, 1], 0.2 GeV ≤ pT ≤ 100 GeV;
• 1000 “noise” tracks, with ϕ uniform in [0, 2π], z of track uniform in [−600 mm, 600 mm],
η uniform in [−1, 1], 0.2 GeV ≤ pT ≤ 0.8 GeV;
• 4 “jets” with 10 tracks each. Jet axis: ϕ uniform in [0, 2π], z uniform in [−100 mm, 100 mm],
η uniform in [−1, 1]; jet width: ∆ϕ ≤ 0.2, ∆θ ≤ 0.2; track momentum in jet: 20 GeV ≤ pT ≤
100 GeV.
pT was generated according to the formula pT = pmin pmax /(pmax − u(pmax − pmin )), with u uniform in [0,1]. The distribution of the transverse momentum of the “primary” tracks is shown in
figure 1. About 20% of the tracks have a transverse momentum above 1 GeV.
3
Vector hits
Vector hits or stubs are short track segments reconstructed from two hits in stacked sensors. A
vector hit contains four track parameters plus their covariance matrix. The track parameters are:
the azimuthal position angle Φ, the longitudinal position z, the polar track angle θ , and the local
track angle β = φ − Φ. The curvature cannot be estimated locally in a stacked module, but can
be reconstructed using the beam profile. Vector hits are reconstructed only for tracks above a
transverse momentum threshold of 1 GeV. The required cuts are determined as follows:
–2–
2012 JINST 7 C09002
• Pixel size: 100 µm(RΦ) × 1 mm(z);
5
10
4
10
3
N
10
2
10
0
10
0
10
20
30
40
50
60
Transverse momentum
70
80
90
100
Figure 1. Distribution of the transverse momentum of the “primary” tracks.
• Select tracks with simulated pT > 1 GeV, thus no “noise” tracks are selected;
• In each layer, select corresponding hit pairs;
• Correct z position for track angle θ : z0 = z − r ∗ cot(θ ) + z̄, where z̄ is the average z position;
• Compute ∆Φ and ∆z0 ;
• Determine cuts by cΦ = 1.05 · max |∆Φ|, cz0 = 1.05 · max |∆z0 |.
The cuts depend on the spread of the tracks in the z-direction, and they have to be wider with
increasing stack separation. Their actual values are shown in figure 2.
The reconstruction of the vector hits then proceeds as follows:
• In each layer, select all hit pairs passing the cuts on Φ and z0 ;
• For each hit pair, estimate θ = arctan(∆r/∆z); estimate the local track direction φ and track
curvature κ using two hits and the beam line (z axis); and fit a helix to obtain (Φ, z, θ , β =
φ − Φ) plus covariance matrix.1
The resolution of Φ and z is determined mainly by the pixel size (or the pitch in a strip sensor);
the resolution of θ , β , κ depends strongly on the stack separation ∆r. This is illustrated in figure 3.
As larger stack separation implies larger cuts in Φ and z, we expect more background vector
hits, i.e. combinations of hits from different tracks. Figure 4 shows that the level of background
hits is still manageable even with the largest stack separation (Run E).
4
Track finding with Kalman filter
Track following with the Kalman filter [2, 3] starts in the outermost layer. For each reconstructed
vector hit (seed), 1. the track is extrapolated to the next inner layer; 2. a search window (±5σ in
1 The
curvature κ is used in the expansion point in the local helix track model, but retains a large error.
–3–
2012 JINST 7 C09002
1
10
7
4
Run A
Run B
Run C
Run D
Run E
6
3.5
3
Cut in z’ [mm]
Cut in Φ [mrad]
5
4
2.5
2
2
1.5
1
2
3
4
5
1
6
1
2
3
Layer
4
5
6
Layer
Figure 2. Size of the cuts in Φ and z for the five choices of stack separation.
0.03
Run A
Run B
Run C
Run D
Run E
0.2
0.16
Run A
Run B
Run C
Run D
Run E
0.025
Run A
Run B
Run C
Run D
Run E
0.14
0.12
0.02
0.1
rms(κ) [m−1]
0.1
rms(β) [rad]
rms(θ) [rad]
0.15
0.015
0.08
0.06
0.01
0.04
0.05
0.005
0.02
0
0
2
4
Layer
6
0
0
2
4
Layer
6
0
0
2
4
6
Layer
Figure 3. Resolution of vector hits for the five choices of stack separation.
Φ and z) is defined; 3. the χ 2 distance to all vector hits in the search window is computed; 4. the
closest vector hit is selected; and 5. the track state is updated. The entire sequence is then repeated
until the innermost layer is reached. Note that the update is restricted to the closest vector hit in
the search window, so that there is no combinatorial explosion and the time required for finding a
track with a given seed is fixed.
–4–
2012 JINST 7 C09002
3
1
Run A
Run B
Run C
Run D
Run E
Run D
20
6
4
2
0
−2
Run E
60
50
Background vector hits/event
8
Background vector hits/event
25
15
10
5
0
0
2
4
Layer
6
−5
40
30
20
10
0
0
2
4
Layer
6
−10
0
2
4
Layer
6
Figure 4. Number of background hits per layer for runs C, D, and E (average value and standard deviation).
In order to asses the performance of the track finder, track candidates (TC) and simulated
tracks (ST) are classified according to the well-defined criteria. The track candidates are assigned
to one of the following classes:
• unTC: unique TC (all hits from the same track);
• mjTC: majority TC (majority of hits from the same track);
• ghTC: ghost TC (no majority of hits from the same track).
The simulated tracks are assigned to one of the following classes:
• unST: ST found uniquely (by a unique track candidate);
• mjST: ST found in majority (by a majority track candidate);
• lsST: ST lost (not found by a unique or majority track candidate).
In many cases a majority track candidate is really unique, because two tracks may produce the
same vector hit.
Table 2 shows the results of the KF track finder for “primary” tracks above the transverse
momentum threshold and for “jet tracks”, averaged over 500 events. For “primary” tracks, the
classification of the track candidates is shown as well. It can be seen that the purity of the track
candidates is excellent. “Jet” tracks have a somewhat lower track finding efficiency, as the probability of picking up a wrong hit is larger in a dense bundle of tracks.
–5–
2012 JINST 7 C09002
Background vector hits/event
Run C
10
Table 2. KF track finder: classification of “primary” and “jet” tracks; track finding efficiency; classification
of track candidates.
Primary
Jet
ST
238.0
238.7
237.7
ST
40
40
40
unST
189.1
234.6
237.2
unST
26.57
37.90
39.64
mjST
0.174
0.158
0.142
mjST
0.070
0.062
0.062
eff
0.795
0.983
0.998
eff
0.666
0.949
0.993
TC
201.0
244.3
246.4
unTC
200.8
244.0
246.2
mjTC
0.178
0.190
0.164
ghTC
0.092
0.058
0.036
Table 3. CHT track finder: classification of “primary” and “jet” tracks; track finding efficiency; classification
of track candidates.
Primary
Jet
5
Run
C
D
E
Run
C
D
E
ST
237.7
239.1
238.3
ST
40
40
40
unST
219.4
229.1
232.1
unST
26.64
31.42
34.39
mjST
16.12
8.79
5.38
mjST
11.62
7.66
5.01
eff
0.991
0.995
0.996
eff
0.956
0.977
0.985
TC
261.3
257.0
257.2
unTC
228.4
236.4
239.5
mjTC
26.76
17.12
14.61
ghTC
6.15
3.45
3.01
Track finding with convergent Hough transform
First, for each vector hit, we compute the azimuth angle ϕ, the polar angle ϑ and the curvature κ
at the point of closest approach to the beam line (z-axis). Next, in each layer, the joint covariance
matrix of (ϕ,ϑ ,κ) is estimated for tracks above the pT threshold. Finally, we search for clusters
in the Hough space (ϕ,ϑ ,κ) . To this end, we find all compatible combinations of hits in layers 5
and 6, and for each of those, we find the best matching vector hits in layers 1–4. The spread of the
clusters depends strongly on the stack separation — larger stack separation improves the resolution
of the parameters in the Hough space and leads to a better cluster separation.
Table 3 shows the results of the CHT track finder for “primary” tracks above the pT threshold
and for “jet tracks”, averaged over 500 events. For “primary” tracks, the classification of the track
candidates is shown as well. Compared to the KF, the purity of the track candidates is considerably
worse. Again, “jet” tracks have a somewhat lower track finding efficiency.
6
Discussion and outlook
With a stack separation of 2 mm in all layers, the track finding efficiency of the KF track finder is
only about 80%. As the KF track finder used in this study proceeds inwards, it is natural to increase
–6–
2012 JINST 7 C09002
Run
A
B
C
Run
A
B
C
References
[1] A. Ryd, The CMS Track Trigger Upgrade for SLHC, in Proceedings of the Vertex 2009 conference,
PoS(VERTEX 2009)040.
[2] R. Frühwirth, Application of Kalman filtering to track and vertex fitting, Nucl. Instrum. Meth. A 262
(1987) 444.
[3] P. Billoir, Progressive track recognition with a Kalman-like fitting procedure, Comput. Phys. Commun.
57 (1989) 390 [INSPIRE].
[4] M. Ohlsson, C. Peterson and A.L. Yuille, Track Finding with Deformable Templates — The Elastic
Arms Approach, Comput. Phys. Commun. 71 (1992) 77.
[5] R. Frühwirth and A. Strandlie, Pattern Recognition and Reconstruction, in Elementary Particles,
Landolt-Börnstein New Series I/21B1, C.W. Fabjan and H. Schopper eds., Springer, Heidelberg
Germany (2011).
–7–
2012 JINST 7 C09002
the stack separation and thereby improve the vector hit resolution in the outer layers. Table 2 shows
that this has the desired effect. In run C, with stack separations of 2/2/4/4/6/6 mm, the efficiency is
very close to 100%, also for the “jet” tracks, and very few tracks are contaminated by extraneous
hits. If the KF track finder would proceed outwards, increasing the stack separation in the outer
layers would not help. This is another example of the close coupling of detector design and track
finding algorithms.
The results of run C allow a direct comparison of the KF track finder and the CHT track finder.
The latter has somewhat lower efficiency, and the track purity is clearly worse. The efficiency can
be raised by further increasing the stack separation, but the number of contaminated tracks stays at
the level of a few percent. There are also more ghost track candidates, but all of these are in the
“jets”, where the chance of confusion is higher. Faking of high pT tracks is thus unlikely.
We conclude that vector hits are useful in track finding if they have substantial direction information, i.e. if the stack separations are not too small. It is therefore important to find an upper
limit of the stack separation that is still compatible with the trigger requirements and the expected
background level. The present work gives some indications but contains too many simplifications
to allow definite statements. It is clear that the simulations have to be made more realistic and that
more methods have to be studied. In particular, a full detector simulation with GEANT4 is indispensable for a meaningful optimization of the layer positions and the stack separations for different
track finding algorithms and geometries.
In the present study, the vector hit reconstruction is an elaborate helix fit, which is certainly too
slow for level 1 reconstruction. If it is replaced by a simple straight line fit, one has to study whether
the loss in precision is acceptable. Concerning the track finding itself, the Hough transform is a
candidate for L1 deployment, particularly if the clustering is implemented in hardware (associative
memory). Some related studies can be found in [6, 7]. The full Kalman filter is probably too slow
for L1, and it would be interesting to see whether it can be replaced by a cellular automaton [8], in
connection with multi-layer perceptrons trained for fast selection of track segments and neighbors.
To sum up, there is the opportunity to design and optimize the detector and the track finding
algorithms at the same time, and one should take advantage of this fortunate situation.
[6] A. Ryd, E. Salvati and Y. Weng, L1 track finding in the long barrel geometry,
https://indico.cern.ch/getFile.py/access?contribId=5&sessionId=2&resId=0&
materialId=slides&confId=197209.
[7] G.Baulieu and S. Viret, L1 track trigger with Barrel+Endcap geometry,
https://indico.cern.ch/getFile.py/access?contribId=6&sessionId=2&resId=0&
materialId=slides&confId=197209.
[8] I. Kisel et al., Cellular automaton and elastic net for event reconstruction in the NEMO-2 experiment,
Nucl. Instrum. Meth. A 387 (1997) 433.
2012 JINST 7 C09002
–8–